Wireless non-radiative energy transfer

ABSTRACT

Described herein are embodiments of a source high-Q resonator, optionally coupled to an energy source, a second high-Q resonator, optionally coupled to an energy drain that may be located a distance from the source resonator. A third high-Q resonator, optionally coupled to an energy drain that may be located a distance from the source resonator. The source resonator and at least one of the second resonator and third resonator may be coupled to transfer electromagnetic energy from said source resonator to said at least one of the second resonator and third resonator.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation and claims the benefit of priorityunder 35 USC §120 to U.S. application Ser. No. 12/639,963, filed Dec.16, 2009, which is a continuation of U.S. application Ser. No.12/553,957, filed Sep. 3, 2009, which is a continuation of U.S.application Ser. No. 11/481,077 filed Jul. 5, 2006, now U.S. Pat. No.7,741,734, which claims priority under 35 USC §119(e) to U.S.provisional application Ser. No. 60/698,442 filed Jul. 12, 2005. Thecontents of the prior applications mentioned above are incorporatedherein by reference in their entirety.

STATEMENT AS TO FEDERALLY FUNDED RESEARCH

This invention was made with government support awarded by the NationalScience Foundation under Grant No. DMR-0213282. The government hascertain rights in this invention.

BACKGROUND OF THE INVENTION

The invention relates to the field of oscillatory resonantelectromagnetic modes, and in particular to oscillatory resonantelectromagnetic modes, with localized slowly evanescent field patterns,for wireless non-radiative energy transfer.

In the early days of electromagnetism, before the electrical-wire gridwas deployed, serious interest and effort was devoted towards thedevelopment of schemes to transport energy over long distanceswirelessly, without any carrier medium. These efforts appear to have metwith little, if any, success. Radiative modes of omnidirectionalantennas, which work very well for information transfer, are notsuitable for such energy transfer, because a vast majority of energy iswasted into free space. Directed radiation modes, using lasers orhighly-directional antennas, can be efficiently used for energytransfer, even for long distances (transfer distanceL_(TRANS))>>L_(DEV), where L_(DEV) is the characteristic size of thedevice), but require existence of an uninterruptible line-of-sight and acomplicated tracking system in the case of mobile objects.

Rapid development of autonomous electronics of recent years (e.g.laptops, cell-phones, house-hold robots, that all typically rely onchemical energy storage) justifies revisiting investigation of thisissue. Today, the existing electrical-wire grid carries energy almosteverywhere; even a medium-range wireless non-radiative energy transferwould be quite useful. One scheme currently used for some importantapplications relies on induction, but it is restricted to veryclose-range (L_(TRANS)<<L_(DEV)) energy transfers.

SUMMARY OF THE INVENTION

According to one aspect of the invention, there is provided anelectromagnetic energy transfer device. The electromagnetic energytransfer device includes a first resonator structure receiving energyfrom an external power supply. The first resonator structure has a firstQ-factor. A second resonator structure is positioned distal from thefirst resonator structure, and supplies useful working power to anexternal load. The second resonator structure has a second Q-factor. Thedistance between the two resonators can be larger than thecharacteristic size of each resonator. Non-radiative energy transferbetween the first resonator structure and the second resonator structureis mediated through coupling of their resonant-field evanescent tails.

According to another aspect of the invention, there is provided a methodof transferring electromagnetic energy. The method includes providing afirst resonator structure receiving energy from an external powersupply. The first resonator structure has a first Q-factor. Also, themethod includes a second resonator structure being positioned distalfrom the first resonator structure, and supplying useful working powerto an external load. The second resonator structure has a secondQ-factor. The distance between the two resonators can be larger than thecharacteristic size of each resonator. Furthermore, the method includestransferring non-radiative energy between the first resonator structureand the second resonator structure through coupling of theirresonant-field evanescent tails.

In another aspect, a method of transferring energy is disclosedincluding the steps of providing a first resonator structure receivingenergy from an external power supply, said first resonator structurehaving a first resonant frequency ω₁, and a first Q-factor Q₁, andcharacteristic size L₁. Providing a second resonator structure beingpositioned distal from said first resonator structure, at closestdistance D, said second resonator structure having a second resonantfrequency ω₂, and a second Q-factor Q₂, and characteristic size L₂,where the two said frequencies ω₁ and ω₂ are close to within thenarrower of the two resonance widths Γ₁, and Γ₂, and transferring energynon-radiatively between said first resonator structure and said secondresonator structure, said energy transfer being mediated throughcoupling of their resonant-field evanescent tails, and the rate ofenergy transfer between said first resonator and said second resonatorbeing denoted by κ, where non-radiative means D is smaller than each ofthe resonant wavelengths λ₁ and λ₂, where c is the propagation speed ofradiation in the surrounding medium.

Embodiments of the method may include any of the following features. Insome embodiments, said resonators have Q₁>100 and Q₂>100, Q₁>200 andQ₂>200, Q₁>500 and Q₂>500, or even Q₁>1000 and Q₂>1000. In some suchembodiments, κ/sqrt(Γ₁*Γ₂) may be greater than 0.2, greater than 0.5,greater than 1, greater than 2, or even grater than 5. In some suchembodiments D/L₂ may be greater than 1, greater than 2, greater than 3,greater than 5.

In another aspect, an energy transfer device is disclosed whichincludes: a first resonator structure receiving energy from an externalpower supply, said first resonator structure having a first resonantfrequency ω₁, and a first Q-factor Q₁, and characteristic size L₁, and asecond resonator structure being positioned distal from said firstresonator structure, at closest distance D, said second resonatorstructure having a second resonant frequency ω₂, and a second Q-factorQ₂, and characteristic size L₂.

The two said frequencies ω₁ and ω₂ are close to within the narrower ofthe two resonance widths Γ₁, and Γ₂. The non-radiative energy transferbetween said first resonator structure and said second resonatorstructure is mediated through coupling of their resonant-fieldevanescent tails, and the rate of energy transfer between said firstresonator and said second resonator is denoted by κ. The non-radiativemeans D is smaller than each of the resonant wavelengths λ₁ and λ₂,where c is the propagation speed of radiation in the surrounding medium.

Embodiments of the device may include any of the following features. Insome embodiments, said resonators have Q₁>100 and Q₂>100, Q₁>200 andQ₂>200, Q₁>500 and Q₂>500, or even Q₁>1000 and Q₂>1000. In some suchembodiments, κ/sqrt(Γ₁*Γ₂) may be greater than 0.2, greater than 0.5,greater than 1, greater than 2, or even grater than 5. In some suchembodiments D/L₂ may be greater than 1, greater than 2, greater than 3,or even greater than 5.

In some embodiments, the resonant field in the device iselectromagnetic.

In some embodiments, the first resonator structure includes a dielectricsphere, where the characteristic size L₁ is the radius of the sphere.

In some embodiments, the first resonator structure includes a metallicsphere, where the characteristic size L₁ is the radius of the sphere.

In some embodiments, the first resonator structure includes ametallodielectric sphere, where the characteristic size L₁ is the radiusof the sphere.

In some embodiments, the first resonator structure includes a plasmonicsphere, where the characteristic size L₁ is the radius of the sphere.

In some embodiments, the first resonator structure includes apolaritonic sphere, where the characteristic size L₁ is the radius ofthe sphere.

In some embodiments, the first resonator structure includes acapacitively-loaded conducting-wire loop, where the characteristic sizeL₁ is the radius of the loop.

In some embodiments, the second resonator structure includes adielectric sphere, where the characteristic size L₂ is the radius of thesphere.

In some embodiments, the second resonator structure includes a metallicsphere where the characteristic size L₂ is the radius of the sphere.

In some embodiments, the second resonator structure includes ametallodielectric sphere where the characteristic size L₂ is the radiusof the sphere.

In some embodiments, the second resonator structure includes a plasmonicsphere where the characteristic size L₂ is the radius of the sphere.

In some embodiments, the second resonator structure includes apolaritonic sphere where the characteristic size L₂ is the radius of thesphere.

In some embodiments, the second resonator structure includes acapacitively-loaded conducting-wire loop where the characteristic sizeL₂ is the radius of the loop.

In some embodiments, the resonant field in the device is acoustic.

It is to be understood that embodiments of the above described methodsand devices may include any of the above listed features, alone or incombination.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram illustrating an exemplary embodiment ofthe invention;

FIG. 2A is a numerical FDTD result for a high-index disk cavity ofradius r along with the electric field; FIG. 2B a numerical FDTD resultfor a medium-distance coupling between two resonant disk cavities:initially, all the energy is in one cavity (left panel); after some timeboth cavities are equally excited (right panel).

FIG. 3 is schematic diagram demonstrating two capacitively-loadedconducting-wire loops;

FIGS. 4A-4B are numerical FDTD results for reduction in radiation-Q ofthe resonant disk cavity due to scattering from extraneous objects;

FIG. 5 is a numerical FDTD result for medium-distance coupling betweentwo resonant disk cavities in the presence of extraneous objects; and

FIGS. 6A-6B are graphs demonstrating efficiencies of converting thesupplied power into useful work (η_(w)), radiation and ohmic loss at thedevice (η_(d)), and the source (η_(s)), and dissipation inside a human(η_(h)), as a function of the coupling-to-loss ratio κ/Γ_(d); in panel(a) Γ_(w) is chosen so as to minimize the energy stored in the device,while in panel (b) Γ_(w) is chosen so as to maximize the efficiencyη_(w) for each κ/Γ_(d).

FIG. 7 is a schematic diagram of a feedback mechanism to correct theresonators exchanging wireless energy for detuning because of the effectof an extraneous object.

FIG. 8 is a schematic diagram of an exemplary system in which a high-Qsource resonator 810 is receiving power from a power supply 820 andwirelessly transfers non-radiative electromagnetic energy to multiplehigh-Q device resonators 830 and 840, each of which is coupled to anenergy drain 850 and 860, respectively.

DETAILED DESCRIPTION OF THE INVENTION

In contrast to the currently existing schemes, the invention providesthe feasibility of using long-lived oscillatory resonant electromagneticmodes, with localized slowly evanescent field patterns, for wirelessnon-radiative energy transfer. The basis of this technique is that twosame-frequency resonant objects tend to couple, while interacting weaklywith other off-resonant environmental objects. The purpose of theinvention is to quantify this mechanism using specific examples, namelyquantitatively address the following questions: up to which distancescan such a scheme be efficient and how sensitive is it to externalperturbations. Detailed theoretical and numerical analysis show that amid-range (L_(TRANS)≈few*L_(DEV)) wireless energy-exchange can actuallybe achieved, while suffering only modest transfer and dissipation ofenergy into other off-resonant objects.

The omnidirectional but stationary (non-lossy) nature of the near fieldmakes this mechanism suitable for mobile wireless receivers. It couldtherefore have a variety of possible applications including for example,placing a source connected to the wired electricity network on theceiling of a factory room, while devices, such as robots, vehicles,computers, or similar, are roaming freely within the room. Otherpossible applications include electric-engine buses, RFIDs, and perhapseven nano-robots. Similarly, in some embodiments multiple sources cantransfer energy to one or more device objects. For example, as explainedat in the paragraph bridging pages 4-5 of U.S. Provisional ApplicationNo. 60/698,442 to which the present application claims benefit and whichis incorporated by reference above, for certain applications havinguneven power transfer to the device object as the distance between thedevice and the source changes, multiple sources can be strategicallyplaced to alleviate the problem, and/or the peak amplitude of the sourcecan be dynamically adjusted.

The range and rate of the inventive wireless energy-transfer scheme arethe first subjects of examination, without considering yet energydrainage from the system for use into work. An appropriate analyticalframework for modeling the exchange of energy between resonant objectsis a weak-coupling approach called “coupled-mode theory”. FIG. 1 is aschematic diagram illustrating a general description of the invention.The invention uses a source and device to perform energy transferring.Both the source 1 and device 2 are resonator structures, and areseparated a distance D from each other. In this arrangement, theelectromagnetic field of the system of source 1 and device 2 isapproximated by F(r,t)≈a₁(t)F₁(r)+a₂(t)F₂(r), whereF_(1,2)(r)=[E_(1,2)(r) H_(1,2)(r)] are the eigenmodes of source 1 anddevice 2 alone, and then the field amplitudes a₁(t) and a₂(t) can beshown to satisfy the “coupled-mode theory”:

$\begin{matrix}{{\frac{\mathbb{d}a_{1}}{\mathbb{d}t} = {{{- {{\mathbb{i}}( {\omega_{1} - {\mathbb{i}\Gamma}_{1}} )}}a_{1}} + {{\mathbb{i}\kappa}_{11}a_{1}} + {{\mathbb{i}\kappa}_{12}a_{2}}}}{{\frac{\mathbb{d}a_{2}}{\mathbb{d}t} = {{{- {{\mathbb{i}}( {\omega_{2} - {\mathbb{i}\Gamma}_{2}} )}}a_{2}} + {{\mathbb{i}\kappa}_{22}a_{2}} + {{\mathbb{i}\kappa}_{21}a_{1}}}},}} & (1)\end{matrix}$where ω_(1,2) are the individual eigen-frequencies, Γ_(1,2) are theresonance widths due to the objects' intrinsic (absorption, radiationetc.) losses, κ_(12,21) are the coupling coefficients, and κ_(11,22)model the shift in the complex frequency of each object due to thepresence of the other.

The approach of Eq. 1 has been shown, on numerous occasions, to providean excellent description of resonant phenomena for objects of similarcomplex eigen-frequencies (namely |ω₁−ω₂|<<|κ_(12,21)| and Γ₁≈Γ₂), whoseresonances are reasonably well defined (namelyΓ_(1,2)&Im{κ_(11,22)}<<|K_(12,21)|) and in the weak coupling limit(namely |κ_(12,21)|<<ω_(1,2)). Coincidentally, these requirements alsoenable optimal operation for energy transfer. Also, Eq. (1) show thatthe energy exchange can be nearly perfect at exact resonance (ω₁=ω₂ andΓ₁=Γ₂), and that the losses are minimal when the “coupling-time” is muchshorter than all “loss-times”. Therefore, the invention requiresresonant modes of high Q=ω/(2Γ) for low intrinsic-loss rates Γ_(1,2),and with evanescent tails significantly longer than the characteristicsizes L₁ and L₂ of the two objects for strong coupling rate |κ_(12,21)|over large distances D, where D is the closest distance between the twoobjects. This is a regime of operation that has not been studiedextensively, since one usually prefers short tails, to minimizeinterference with nearby devices.

Objects of nearly infinite extent, such as dielectric waveguides, cansupport guided modes whose evanescent tails are decaying exponentiallyin the direction away from the object, slowly if tuned close to cutoff,and can have nearly infinite Q. To implement the inventiveenergy-transfer scheme, such geometries might be suitable for certainapplications, but usually finite objects, namely ones that aretopologically surrounded everywhere by air, are more appropriate.

Unfortunately, objects of finite extent cannot support electromagneticstates that are exponentially decaying in all directions in air, sincein free space: {right arrow over (k)}=ω²/c². Because of this, one canshow that they cannot support states of infinite Q. However, verylong-lived (so-called “high-Q”) states can be found, whose tails displaythe needed exponential-like decay away from the resonant object overlong enough distances before they turn oscillatory (radiative). Thelimiting surface, where this change in the field behavior happens, iscalled the “radiation caustic”, and, for the wireless energy-transferscheme to be based on the near field rather than the far/radiationfield, the distance between the coupled objects must be such that onelies within the radiation caustic of the other.

The invention is very general and any type of resonant structuresatisfying the above requirements can be used for its implementation. Asexamples and for definiteness, one can choose to work with twowell-known, but quite different electromagnetic resonant systems:dielectric disks and capacitively-loaded conducting-wire loops. Evenwithout optimization, and despite their simplicity, both will be shownto exhibit fairly good performance. Their difference lies mostly in thefrequency range of applicability due to practical considerations, forexample, in the optical regime dielectrics prevail, since conductivematerials are highly lossy.

Consider a 2D dielectric disk cavity of radius r and permittivity &surrounded by air that supports high-Q whispering-gallery modes, asshown in FIG. 2A. Such a cavity is studied using both analyticalmodeling, such as separation of variables in cylindrical coordinates andapplication of boundary conditions, and detailed numericalfinite-difference-time-domain (FDTD) simulations with a resolution of 30pts/r. Note that the physics of the 3D case should not be significantlydifferent, while the analytical complexity and numerical requirementswould be immensely increased. The results of the two methods for thecomplex eigen-frequencies and the field patterns of the so-called“leaky” eigenmodes are in an excellent agreement with each other for avariety of geometries and parameters of interest.

The radial modal decay length, which determines the coupling strengthκ≡|κ₂₁|=|κ₁₂|, is on the order of the wavelength, therefore, fornear-field coupling to take place between cavities whose distance ismuch larger than their size, one needs subwavelength-sized resonantobjects (r<<λ). High-radiation-Q and long-tailed subwavelengthresonances can be achieved, when the dielectric permittivity ∈ is aslarge as practically possible and the azimuthal field variations (ofprincipal number m) are slow (namely m is small).

One such TE-polarized dielectric-cavity mode, which has the favorablecharacteristics Q_(rad)=1992 and λ/r=20 using ∈=147.7 and m=2, is shownin FIG. 2A, and will be the “test” cavity 18 for all subsequentcalculations for this class of resonant objects. Another example of asuitable cavity has Q_(rad)=9100 and λ/r=10 using ∈=65.61 and m=3. Thesevalues of ∈ might at first seem unrealistically large. However, not onlyare there in the microwave regime (appropriate for meter-range couplingapplications) many materials that have both reasonably high enoughdielectric constants and low losses, for example, Titania: ∈≈96,Im{∈}/∈≈10⁻³; Barium tetratitanate: ∈37, Im{∈}/∈≈10⁻⁴; Lithiumtantalite: ∈≈40, Im{∈}/∈10⁻⁴; etc.), but also ∈ could instead signifythe effective index of other known subwavelength (λ/r>>1) surface-wavesystems, such as surface-plasmon modes on surfaces of metal-like(negative-∈) materials or metallodielectric photonic crystals.

With regards to material absorption, typical loss tangents in themicrowave (e.g. those listed for the materials above) suggest thatQ_(abs)˜∈/Im{∈}10000. Combining the effects of radiation and absorption,the above analysis implies that for a properly designed resonantdevice-object d a value of Q_(d)˜2000 should be achievable. Note though,that the resonant source s will in practice often be immobile, and therestrictions on its allowed geometry and size will typically be muchless stringent than the restrictions on the design of the device;therefore, it is reasonable to assume that the radiative losses can bedesigned to be negligible allowing for Q_(s)˜10000, limited only byabsorption.

To calculate now the achievable rate of energy transfer, one can placetwo of the cavities 20, 22 at distance D between their centers, as shownin FIG. 2B. The normal modes of the combined system are then an even andan odd superposition of the initial modes and their frequencies aresplit by the coupling coefficient κ, which we want to calculate.Analytically, coupled-mode theory gives for dielectric objectsκ₁₂=ω₂/2·∫d³rE₁*(r)E₂(r)∈₁(r)/∫d³r|E₁(r)|²∈(r), where ∈_(1,2)(r) denotethe dielectric functions of only object 1 alone or 2 alone excluding thebackground dielectric (free space) and ∈(r) the dielectric function ofthe entire space with both objects present. Numerically, one can find κusing FDTD simulations either by exciting one of the cavities andcalculating the energy-transfer time to the other or by determining thesplit normal-mode frequencies. For the “test” disk cavity the radiusr_(C) of the radiation caustic is r_(C)≈11r, and for non-radiativecoupling D<r_(C), therefore here one can choose D/r=10, 7, 5, 3. Then,for the mode of FIG. 3, which is odd with respect to the line thatconnects the two cavities, the analytical predictions are ω/2κ=1602,771, 298, 48, while the numerical predictions are ω/2κ=1717, 770, 298,47 respectively, so the two methods agree well. The radiation fields ofthe two initial cavity modes interfere constructively or destructivelydepending on their relative phases and amplitudes, leading to increasedor decreased net radiation loss respectively, therefore for any cavitydistance the even and odd normal modes have Qs that are one larger andone smaller than the initial single-cavity Q=1992 (a phenomenon notcaptured by coupled-mode theory), but in a way that the average Γ isalways approximately Γ≈ω/2Q. Therefore, the correspondingcoupling-to-loss ratios are κ/Γ=1.16, 2.59, 6.68, 42.49, and althoughthey do not fall in the ideal operating regime κ/Γ>>1, the achievedvalues are still large enough to be useful for applications.

Consider a loop 10 or 12 of N coils of radius r of conducting wire withcircular cross-section of radius a surrounded by air, as shown in FIG.3. This wire has inductance L=μ_(o)N²r[ln(8r/a)−2], where μ_(o) is themagnetic permeability of free space, so connecting it to a capacitance Cwill make the loop resonant at frequency ω=1/√{square root over (LC)}.The nature of the resonance lies in the periodic exchange of energy fromthe electric field inside the capacitor due to the voltage across it tothe magnetic field in free space due to the current in the wire. Lossesin this resonant system consist of ohmic loss inside the wire andradiative loss into free space.

For non-radiative coupling one should use the near-field region, whoseextent is set roughly by the wavelength λ, therefore the preferableoperating regime is that where the loop is small (r<<λ). In this limit,the resistances associated with the two loss channels are respectivelyR_(ohm)=√{square root over (μ_(o)ρω/2)}·Nr/a andR_(rad)=π/6·η_(o)N²)ωr/c)⁴, where ρ is the resistivity of the wirematerial and η_(o)≈120πΩ is the impedance of free space. The qualityfactor of such a resonance is then Q=ωL/(R_(ohm)+R_(rad)) and is highestfor some frequency determined by the system parameters: at lowerfrequencies it is dominated by ohmic loss and at higher frequencies byradiation.

To get a rough estimate in the microwave, one can use one coil (N=1) ofcopper (ρ=1.69·10⁻⁸ Ωm) wire and then for r=1 cm and a=1 mm, appropriatefor example for a cell phone, the quality factor peaks to Q=1225 atf=380 MHz, for r=30 cm and a=2 mm for a laptop or a household robotQ=1103 at f=17 MHz, while for r=1 m and a=4 mm (that could be a sourceloop on a room ceiling) Q=1315 at f=5 MHz. So in general, expectedquality factors are Q≈1000-1500 at λ/r≈50-80, namely suitable fornear-field coupling.

The rate for energy transfer between two loops 10 and 12 at distance Dbetween their centers, as shown in FIG. 3, is given by κ₁₂=ωM/2√{squareroot over (L₁L₂)}, where M is the mutual inductance of the two loops 10and 12. In the limit r<<D<<λ one can use the quasi-static resultM=π/4·μ_(o)N₁N₂(r₁r₂)²/D³, which means that ω/2κ˜(D/√{square root over(r₁r₂)})³. For example, by choosing again D/r=10, 8, 6 one can get fortwo loops of r=1 cm, same as used before, that ω/2κ=3033, 1553, 655respectively, for the r=30 cm that ω₁2K=7131, 3651, 1540, and for ther=1 m that ω₁2K=6481, 3318, 1400. The corresponding coupling-to-lossratios peak at the frequency where peaks the single-loop Q and areκ/Γ=0.4, 0.79, 1.97 and 0.15, 0.3, 0.72 and 0.2, 0.4, 0.94 for the threeloop-kinds and distances. An example of dissimilar loops is that of ar=1 m (source on the ceiling) loop and a r=30 cm (household robot on thefloor) loop at a distance D=3 m (room height) apart, for whichκ/√{square root over (Γ₁Γ₂)}=0.88 peaks at f=6.4 MHz, in between thepeaks of the individual Q's. Again, these values are not in the optimalregime κ/Γ>>1, but will be shown to be sufficient.

It is important to appreciate the difference between this inductivescheme and the already used close-range inductive schemes for energytransfer in that those schemes are non-resonant. Using coupled-modetheory it is easy to show that, keeping the geometry and the energystored at the source fixed, the presently proposed resonant-couplinginductive mechanism allows for Q approximately 1000 times more powerdelivered for work at the device than the traditional non-resonantmechanism, and this is why mid-range energy transfer is now possible.Capacitively-loaded conductive loops are actually being widely used asresonant antennas (for example in cell phones), but those operate in thefar-field regime with r/λ˜1, and the radiation Q's are intentionallydesigned to be small to make the antenna efficient, so they are notappropriate for energy transfer.

Clearly, the success of the inventive resonance-based wirelessenergy-transfer scheme depends strongly on the robustness of theobjects' resonances. Therefore, their sensitivity to the near presenceof random non-resonant extraneous objects is another aspect of theproposed scheme that requires analysis. The interaction of an extraneousobject with a resonant object can be obtained by a modification of thecoupled-mode-theory model in Eq. (1), since the extraneous object eitherdoes not have a well-defined resonance or is far-off-resonance, theenergy exchange between the resonant and extraneous objects is minimal,so the term κ₁₂ in Eq. (1) can be dropped. The appropriate analyticalmodel for the field amplitude in the resonant object a₁ (t) becomes:

$\begin{matrix}{\frac{\mathbb{d}a_{1}}{\mathbb{d}t} = {{{- {{\mathbb{i}}( {\omega_{1} - {\mathbb{i}\Gamma}_{1}} )}}a_{1}} + {{\mathbb{i}\kappa}_{11}a_{1}}}} & (2)\end{matrix}$

Namely, the effect of the extraneous object is just a perturbation onthe resonance of the resonant object and it is twofold: First, it shiftsits resonant frequency through the real part of κ₁₁ thus detuning itfrom other resonant objects. As shown in FIG. 7, this is a problem thatcan be fixed rather easily by applying a feedback mechanism 710 to everydevice (e.g., device resonators 720 and 730) that corrects itsfrequency, such as through small changes in geometry, and matches it tothat of the source resonator 740. Second, it forces the resonant objectto lose modal energy due to scattering into radiation from theextraneous object through the induced polarization or currents in it,and due to material absorption in the extraneous object through theimaginary part of κ₁₁. This reduction in Q can be a detrimental effectto the functionality of the energy-transfer scheme, because it cannot beremedied, so its magnitude must be quantified.

In the first example of resonant objects that have been considered, theclass of dielectric disks, small, low-index, low-material-loss orfar-away stray objects will induce small scattering and absorption. Toexamine realistic cases that are more dangerous for reduction in Q, onecan therefore place the “test” dielectric disk cavity 40 close to: a)another off-resonance object 42, such as a human being, of largeRe{∈}=49 and Im{∈}=16 and of same size but different shape, as shown inFIG. 4A; and b) a roughened surface 46, such as a wall, of large extentbut of small Re{∈}=2.5 and Im{∈}=0.05, as shown in FIG. 4B.

Analytically, for objects that interact with a small perturbation thereduced value of radiation-Q due to scattering could be estimated usingthe polarization ∫d³r|P_(X1)(r)|²∞∫d³r|E₁(r)·Re{∈_(X)(r)}|² induced bythe resonant cavity 1 inside the extraneous object X=42 or roughenedsurface X=46. Since in the examined cases either the refractive index orthe size of the extraneous objects is large, these first-orderperturbation-theory results would not be accurate enough, thus one canonly rely on numerical FDTD simulations. The absorption-Q inside theseobjects can be estimated through{κ₁₁}=ω₁/2·∫d³r|(r)|²Im{∈_(X)(r)}/∫d³r|E₁(r)²∈(r).

Using these methods, for distances D/r=10, 7, 5, 3 between the cavityand extraneous-object centers one can find that Q_(rad)=1992 isrespectively reduced to Q_(rad)=1988, 1258, 702, 226, and that theabsorption rate inside the object is Q_(abs)=312530, 86980, 21864, 1662,namely the resonance of the cavity is not detrimentally disturbed fromhigh-index and/or high-loss extraneous objects, unless the (possiblymobile) object comes very close to the cavity. For distances D/r=10, 7,5, 3, 0 of the cavity to the roughened surface we find respectivelyQ_(rad)=2101, 2257, 1760, 1110, 572, and Q_(abs)>4000, namely theinfluence on the initial resonant mode is acceptably low, even in theextreme case when the cavity is embedded on the surface. Note that aclose proximity of metallic objects could also significantly scatter theresonant field, but one can assume for simplicity that such objects arenot present.

Imagine now a combined system where a resonant source-object s is usedto wirelessly transfer energy to a resonant device-object d but there isan off-resonance extraneous-object e present. One can see that thestrength of all extrinsic loss mechanisms from e is determined by|E_(s)(r_(e))|², by the square of the small amplitude of the tails ofthe resonant source, evaluated at the position r_(e) of the extraneousobject. In contrast, the coefficient of resonant coupling of energy fromthe source to the device is determined by the same-order tail amplitude|E_(s)(r_(d))| evaluated at the position r_(d) of the device, but thistime it is not squared! Therefore, for equal distances of the source tothe device and to the extraneous object, the coupling time for energyexchange with the device is much shorter than the time needed for thelosses inside the extraneous object to accumulate, especially if theamplitude of the resonant field has an exponential-like decay away fromthe source. One could actually optimize the performance by designing thesystem so that the desired coupling is achieved with smaller tails atthe source and longer at the device, so that interference to the sourcefrom the other objects is minimal.

The above concepts can be verified in the case of dielectric diskcavities by a simulation that combines FIGS. 2A-2B and 4A-4B, namelythat of two (source-device) “test” cavities 50 placed 10r apart, in thepresence of a same-size extraneous object 52 of ∈=49 between them, andat a distance 5r from a large roughened surface 56 of ∈=2.5, as shown inFIG. 5. Then, the original values of Q=1992, ω/2κ=1717 (and thusκ/Γ=1.16) deteriorate to Q=765, ω/2κ=965 (and thus κ/Γ=0.79). Thischange is acceptably small, considering the extent of the consideredexternal perturbation, and, since the system design has not beenoptimized, the final value of coupling-to-loss ratio is promising thatthis scheme can be useful for energy transfer.

In the second example of resonant objects being considered, theconducting-wire loops, the influence of extraneous objects on theresonances is nearly absent. The reason for this is that, in thequasi-static regime of operation (r<<λ) that is being considered, thenear field in the air region surrounding the loop is predominantlymagnetic, since the electric field is localized inside the capacitor.Therefore, extraneous objects that could interact with this field andact as a perturbation to the resonance are those having significantmagnetic properties (magnetic permeability Re{μ}>1 or magnetic lossIm{μ}>0). Since almost all common materials are non-magnetic, theyrespond to magnetic fields in the same way as free space, and thus willnot disturb the resonance of a conducting-wire loop. The onlyperturbation that is expected to affect these resonances is a closeproximity of large metallic structures.

An extremely important implication of the above fact relates to safetyconsiderations for human beings. Humans are also non-magnetic and cansustain strong magnetic fields without undergoing any risk. This isclearly an advantage of this class of resonant systems for manyreal-world applications. On the other hand, dielectric systems of high(effective) index have the advantages that their efficiencies seem to behigher, judging from the larger achieved values of κ/Γ, and that theyare also applicable to much smaller length-scales, as mentioned before.

Consider now again the combined system of resonant source s and device din the presence of a human h and a wall, and now let us study theefficiency of this resonance-based energy-transfer scheme, when energyis being drained from the device for use into operational work. One canuse the parameters found before: for dielectric disks,absorption-dominated loss at the source Q_(s)˜10⁴, radiation-dominatedloss at the device Q_(d)˜10³ (which includes scattering from the humanand the wall), absorption of the source- and device-energy at the humanQ_(s-h), Q_(d-h)˜10⁴-10⁵ depending on his/her not-very-close distancefrom the objects, and negligible absorption loss in the wall; forconducting-wire loops, Q_(s)˜Q_(d)˜10³, and perturbations from the humanand the wall are negligible. With corresponding loss-rates Γ=ω/2Q,distance-dependent coupling κ, and the rate at which working power isextracted Γ_(w), the coupled-mode-theory equation for the devicefield-amplitude is

$\begin{matrix}{\frac{\mathbb{d}a_{d}}{\mathbb{d}t} = {{{- {{\mathbb{i}}( {\omega - {\mathbb{i}\Gamma}_{d}} )}}a_{d}} + {{\mathbb{i}\kappa}\; a_{\; s}} - {\Gamma_{d - h}a_{d}} - {\Gamma_{w}{a_{d}.}}}} & (3)\end{matrix}$

Different temporal schemes can be used to extract power from the deviceand their efficiencies exhibit different dependence on the combinedsystem parameters. Here, one can assume steady state, such that thefield amplitude inside the source is maintained constant, namelya_(s)(t)=A_(s)e^(−iωt), so then the field amplitude inside the device isa_(d)(t)=A_(d)e^(−iωt) with A_(d)=iκ/(Γ_(d-h)+Γ_(d-h)+F)A_(s).Therefore, the power lost at the source is P_(s)=2Γ_(s)|A_(s)|², at thedevice it is P_(d)=2Γ_(d)|A_(d)|², the power absorbed at the human isP_(h=)2Γ_(s-h)|A_(s)|²+2Γ_(d-h)|A_(d)|², and the useful extracted poweris P_(w)=2Γ_(w)|A_(d)|². From energy conservation, the total powerentering the system is P_(total)=P_(s)+P_(d)+P_(h)+P_(w). Denote thetotal loss-rates Γ_(s) ^(tot)=Γ_(s)+Γ_(s-h) and Γ_(d)^(tot)=Γ_(d)+Γ_(d-h). Depending on the targeted application, thework-drainage rate should be chosen either Γ_(w)=Γ_(d) ^(tot) tominimize the required energy stored in the resonant objects orΓ_(w)=Γ_(d) ^(tot)√{square root over (1+κ²/Γ_(s) ^(tot)Γ_(s)^(tot))}>Γ_(d) ^(tot) such that the ratio of useful-to-lost powers,namely the efficiency η_(w)=P_(w)/P_(total), is maximized for some valueof κ. The efficiencies η for the two different choices are shown inFIGS. 6A and 6B respectively, as a function of the κ/Γ_(d)figure-of-merit which in turn depends on the source-device distance.

FIGS. 6A-6B show that for the system of dielectric disks and the choiceof optimized efficiency, the efficiency can be large, e.g., at least40%. The dissipation of energy inside the human is small enough, lessthan 5%, for values κ/Γ_(d)>1 and Q_(h)>10⁵, namely for medium-rangesource-device distances (D_(d)/r<10) and most human-source/devicedistances (D_(h)/r>8). For example, for D_(d)/r=10 and D_(h)/r=8, if 10W must be delivered to the load, then, from FIG. 6B, ˜0.4 W will bedissipated inside the human, ˜4 W will be absorbed inside the source,and ˜2.6 W will be radiated to free space. For the system ofconducting-wire loops, the achieved efficiency is smaller, ˜20% forκ/Γ_(d)≈1, but the significant advantage is that there is no dissipationof energy inside the human, as explained earlier.

Even better performance should be achievable through optimization of theresonant object designs. Also, by exploiting the earlier mentionedinterference effects between the radiation fields of the coupledobjects, such as continuous-wave operation at the frequency of thenormal mode that has the larger radiation-Q, one could further improvethe overall system functionality. Thus the inventive wirelessenergy-transfer scheme is promising for many modern applications.Although all considerations have been for a static geometry, all theresults can be applied directly for the dynamic geometries of mobileobjects, since the energy-transfer time κ⁻¹˜1 μs, which is much shorterthan any timescale associated with motions of macroscopic objects.

The invention provides a resonance-based scheme for mid-range wirelessnon-radiative energy transfer. Analyses of very simple implementationgeometries provide encouraging performance characteristics for thepotential applicability of the proposed mechanism. For example, in themacroscopic world, this scheme could be used to deliver power to robotsand/or computers in a factory room, or electric buses on a highway(source-cavity would in this case be a “pipe” running above thehighway). In the microscopic world, where much smaller wavelengths wouldbe used and smaller powers are needed, one could use it to implementoptical inter-connects for CMOS electronics or else to transfer energyto autonomous nano-objects, without worrying much about the relativealignment between the sources and the devices; energy-transfer distancecould be even longer compared to the objects' size, since Im{∈(ω)} ofdielectric materials can be much lower at the required opticalfrequencies than it is at microwave frequencies. Accordingly, FIG. 8 isa schematic drawing showing an exemplary system in which a high-Q sourceresonator 810 is receiving power from a power supply 820 and wirelesslytransfers non-radiative electromagnetic energy to multiple high-Q deviceresonators 830 and 840, each of which is coupled to an energy drain 850and 860, respectively.

As a venue of future scientific research, different material systemsshould be investigated for enhanced performance or different range ofapplicability. For example, it might be possible to significantlyimprove performance by exploring plasmonic systems. These systems canoften have spatial variations of fields on their surface that are muchshorter than the free-space wavelength, and it is precisely this featurethat enables the required decoupling of the scales: the resonant objectcan be significantly smaller than the exponential-like tails of itsfield. Furthermore, one should also investigate using acousticresonances for applications in which source and device are connected viaa common condensed-matter object.

Although the present invention has been shown and described with respectto several preferred embodiments thereof, various changes, omissions andadditions to the form and detail thereof, may be made therein, withoutdeparting from the spirit and scope of the invention.

What is claimed is:
 1. A wireless power system comprising: a sourcemodule comprising a source resonator and a power supply coupled to thesource resonator to provide power to the source resonator, the sourceresonator having a resonant frequency ω₁, an intrinsic loss rate Γ₁, andcapable of storing electromagnetic energy with an intrinsic qualityfactor Q₁=ω₁/(2Γ₁), the source resonator comprising at least one loop ofconductive material and further comprising a capacitance; and a portableelectronic device having a device module comprising a device resonatorand a load coupled to the device resonator to receive power from thedevice resonator and provide power to the portable electronic device,the device resonator having a resonant frequency ω₂, an intrinsic lossrate Γ₂, and capable of storing electromagnetic energy with an intrinsicquality factor Q₂=ω₂/(2Γ₂), the device resonator comprising at least oneloop of conductive material and further comprising a capacitance,wherein the device module is spaced from the source module andconfigured to move freely relative to the source module over a range ofdistances D between the source module and the device module, wherein thesource resonator and the device resonator are configured to resonantlyand wirelessly couple electromagnetic power from the source resonator tothe device resonator using non-radiative electromagnetic inductionhaving an energy transfer rate κ, and wherein the intrinsic loss ratessatisfy κ/√{square root over (Γ₁Γ₂)}>2 over the range of distances D,and wherein f₁=ω₁/(2π) and f₂=ω₂/(2π), and f₁ and f₂, are between about5 MHz and 380 MHz, and each intrinsic loss rate comprises a resistivecomponent and a radiative component.
 2. The wireless power system ofclaim 1, wherein the power provided to the load from the deviceresonator defines a work drainage rate Γ_(w), and wherein the workdrainage rate Γ_(w) is configured to be dynamically set as a function ofthe energy transfer rate κ between the first and second resonators asthe device module is moveable relative to the source module over therange of distances D.
 3. The wireless power system of claim 2, whereinthe work drainage rate Γ_(w) is configured to be dynamically set suchthat the ratio of useful-to-lost power is maximized as a function of theenergy transfer rate κ over the range of distances D.
 4. The wirelesspower system of claim 2, wherein the work drainage rate Γ_(w) isconfigured to be dynamically set such that Γ_(w)=Γ₂√{square root over(1+(κ²/Γ₁·Γ₂))} as a function of the energy transfer rate κ over therange of distances D.
 5. The wireless power system of claim 1, whereinthe power provided to the load from the device resonator defines a workdrainage rate Γ_(w), and wherein the work drainage rate Γ_(w) isconfigured to be set such that Γ_(w)=Γ₂√{square root over(1+(κ²/Γ₁·Γ₂))} for some value of the energy transfer rate κ in therange of distances D as the device module is moveable relative to thesource module over the range of distances D.
 6. The wireless powersystem of claim 1, wherein the power provided to the load from thedevice resonator defines a work drainage rate Γ_(w), and wherein thework drainage rate Γ_(w) is configured to be set such that the ratio ofuseful-to-lost power is maximized for some value of the energy transferrate κ in the range of distances D as the device module is moveablerelative to the source module over the range of distances D.
 7. Thewireless power system of claim 1, wherein the portable electronic deviceis a cell phone, a computer, or a robot.
 8. The wireless power system ofclaim 1, wherein Q₁>100.
 9. The wireless power system of claim 8,wherein Q₂>100.
 10. The wireless power system of claim 1, wherein thesource resonator has a characteristic size L₁ and the device resonatorhas a different characteristic size L₂.
 11. The wireless power system ofclaim 10, wherein the intrinsic quality factors satisfy √{square rootover (Q₁Q₂)}>100.
 12. The wireless power system of claim 1, wherein therange of distances D includes D=6 cm.
 13. The wireless power system ofclaim 1, wherein the range of distances D includes D=8 cm.
 14. Thewireless power system of claim 1, wherein the range of distances Dincludes D=10 cm.
 15. The wireless power system of claim 1, wherein thenon-radiative electromagnetic induction means that the distances D areless than the wavelengths corresponding to the resonant frequencies ω₁and ω₂.
 16. The wireless power system of claim 1, further comprising asecond portable electronic device having a second device modulecomprising a second device resonator and a second load coupled to thesecond device resonator to receive power from the second deviceresonator and provide power to the second portable electronic device,and wherein the second device module is spaced from the source moduleand configured to move freely relative to the source module and whereinthe source resonator and the second device resonator are configured toresonantly and wirelessly couple electromagnetic power from the sourceresonator to the second device resonator using non-radiativeelectromagnetic induction.
 17. The wireless power system of claim 1,further comprising a feedback mechanism coupled to at least one of theresonators to correct for detuning of the resonant frequencies.
 18. Asource module for a wireless power system including a portableelectronic device having a device module having a device resonator and aload coupled to the device resonator to receive power from the deviceresonator and provide power to the portable electronic device, thedevice resonator having a resonant frequency ω₂, an intrinsic loss rateΓ₂, and capable of storing electromagnetic energy with an intrinsicquality factor Q₂=ω₂/(2Γ₂), the source module comprising: a sourceresonator and a power supply coupled to the source resonator to providepower to the source resonator, the source resonator having a resonantfrequency ω₁, an intrinsic loss rate Γ₁, and capable of storingelectromagnetic energy with an intrinsic quality factor Q₁=ω₁/(2Γ₁)>100,the source resonator comprising at least one loop of conductive materialand further comprising a capacitance, wherein the source module isconfigured to be spaced from the device module and the device module canmove freely relative to the source module over a range of distances Dbetween the source module and the device module, wherein the sourceresonator is configured to resonantly and wirelessly provideelectromagnetic power to the device resonator using non-radiativeelectromagnetic induction having an energy transfer rate κ, and whereinthe intrinsic loss rates satisfy κ/√{square root over (Γ₁Γ₂)}>2 over therange of distances D, and wherein f₁=ω₁/(2π) and f₂=ω₂/(2π), and f₁ andf₂, are between about 5 MHz and 380 MHz, and each intrinsic loss ratecomprises a resistive component and a radiative component.
 19. Thesource module of claim 18, wherein Q₁>200.
 20. A device module for usein a wireless power system for providing power to a portable electronicdevice, the wireless power system including a source module having asource resonator and a power supply coupled to the source resonator toprovide power to the source resonator, the source resonator having aresonant frequency φ₁, an intrinsic loss rate Γ₁, and capable of storingelectromagnetic energy with an intrinsic quality factor Q₁=ω₁/(2Γ₁), thedevice module comprising: a device module comprising a device resonatorand a load coupled to the device resonator to receive power from thedevice resonator and provide power to the portable electronic device,the device resonator having a resonant frequency ω₂, an intrinsic lossrate Γ₂, and capable of storing electromagnetic energy with an intrinsicquality factor Q₂=ω₂/(2Γ₂), the device resonator comprising at least oneloop of conductive material and further comprising a capacitance,wherein the device module is configured to be spaced from the sourcemodule and move freely relative to the source module over a range ofdistances D between the source module and the device module, wherein thedevice resonator is configured to resonantly and wirelessly receiveelectromagnetic power from the source resonator using non-radiativeelectromagnetic induction having an energy transfer rate κ, wherein theintrinsic loss rates satisfy κ/√{square root over (Γ₁Γ₂)}>2 over therange of distances D, and wherein f₁=ω₁/(2π) and f₂=ω₂/(2π), and f₁ andf₂, are between about 5 MHz and 380 MHz, and each intrinsic loss ratecomprises a resistive component and a radiative component, and whereinthe power provided to the load from the device resonator defines a workdrainage rate Γ_(w), and wherein the work drainage rate Γ_(w) isconfigured to be dynamically set as a function of the energy transferrate κ between the first and second resonators as the device module ismovable relative to the source module over the range of distances D. 21.A device module for use in a wireless power system for providing powerto a portable electronic device, the wireless power system including asource module having a source resonator and a power supply coupled tothe source resonator to provide power to the source resonator, thesource resonator having a resonant frequency ω₁, an intrinsic loss rateΓ₁, and capable of storing electromagnetic energy with an intrinsicquality factor Q₁=ω₁/(2Γ₁), the device module comprising: a devicemodule comprising a device resonator and a load coupled to the deviceresonator to receive power from the device resonator and provide powerto the portable electronic device, the device resonator having aresonant frequency ω₂, an intrinsic loss rate Γ₂, and capable of storingelectromagnetic energy with an intrinsic quality factor Q₂=ω₂/(2Γ₂), thedevice resonator comprising at least one loop of conductive material andfurther comprising a capacitance, wherein the device module isconfigured to be spaced from the source module and move freely relativeto the source module over a range of distances D between the sourcemodule and the device module, wherein the device resonator is configuredto resonantly and wirelessly receive electromagnetic power from thesource resonator using non-radiative electromagnetic induction having anenergy transfer rate κ, wherein the intrinsic loss rates satisfyκ/√{square root over (Γ₁Γ₂)}>2 over the range of distances D, andwherein f₁=ω₁/(2π) and f₂=ω₂/(2π), and f₁ and f₂, are between about 5MHz and 380 MHz, and each intrinsic loss rate comprises a resistivecomponent and a radiative component, and wherein the power provided tothe load from the device resonator defines a work drainage rate Γ_(w),and wherein the work drainage rate Γ_(w) is configured to be set suchthat the ratio of useful-to-lost power is maximized for some value ofthe energy transfer rate κ in the range of distances D as the devicemodule is moveable relative to the source module over the range ofdistances D.
 22. The device module of claim 21, wherein the workdrainage rate Γ_(w) is configured to be set such that Γ_(w)=Γ₂√{squareroot over (1+(κ²/Γ₁·Γ₂))} for said value of the energy transfer rate κin the range of distances D as the device module is moveable relative tothe source module over the range of distances D.
 23. A method forproviding power wirelessly to a portable electronic device, wherein theportable electronic device is configured for use with a source modulehaving a source resonator and a power supply coupled to the sourceresonator to provide power to the source resonator, the source resonatorhaving a resonant frequency ω₁, an intrinsic loss rate Γ₁, and capableof storing electromagnetic energy with an intrinsic quality factorQ₁=ω₁/(2Γ₁), the method comprising: providing the portable electronicdevice with a device resonator and a load coupled to the deviceresonator to receive power from the device resonator and provide powerto the portable electronic device, the device resonator having aresonant frequency ω₂, an intrinsic loss rate Γ₂, and capable of storingelectromagnetic energy with an intrinsic quality factor Q₂=ω₂/(2Γ₂), thedevice resonator comprising at least one loop of conductive material andfurther comprising a capacitance, wherein the device module is spacedfrom the source module and configured to move freely relative to thesource module over a range of distances D between the source module andthe device module; and resonantly and wirelessly receivingelectromagnetic power at the device resonator from the source resonatorusing non-radiative electromagnetic induction having an energy transferrate κ, wherein the intrinsic loss rates satisfy κ/√{square root over(Γ₁F₂)}>1 over the range of distances D, wherein f₁=ω₁/(2π) andf₂=ω₂/(2π), and f₁ and f₂, are between about 5 MHz and 380 MHz, andwherein each intrinsic loss rate comprises a resistive component and aradiative component.
 24. The method of claim 23, further comprising:providing power to the load in the portable electronic device from thedevice resonator, wherein the power provided to the load from the deviceresonator defines a work drainage rate Γ_(w), and wherein the workdrainage rate Γ_(w) is dynamically set as a function of the energytransfer rate κ between the first and second resonators as the devicemodule moves relative to the source module over the range of distancesD.
 25. The method of claim 24, wherein the work drainage rate Γ_(w) isdynamically set such that the ratio of useful-to-lost power is maximizedas a function of the energy transfer rate κ over the range of distancesD.
 26. The method of claim 24, wherein the work drainage rate Γ_(w) isdynamically set such that Γ_(w)=Γ₂/√{square root over (1+(κ²/Γ₁·Γ₂))} asa function of the energy transfer rate κ over the range of distances D.27. The method of claim 23, further comprising: providing power to theload in the portable electronic device from the device resonator,wherein the power provided to the load from the device resonator definesa work drainage rate Γ_(w), and wherein the work drainage rate Γ_(w) isset such that Γ_(w)=Γ₂/√{square root over (1+(κ²/Γ₁·Γ₂))} for some valueof the energy transfer rate κ in the range of distances D as the devicemodule is moveable relative to the source module over the range ofdistances D.
 28. The method of claim 23, further comprising: providingpower to the load in the portable electronic device from the deviceresonator, wherein the power provided to the load from the deviceresonator defines a work drainage rate Γ_(w), and wherein the workdrainage rate Γ_(w) is set such that the ratio of useful-to-lost poweris maximized for some value of the energy transfer rate κ in the rangeof distances D as the device module is moveable relative to the sourcemodule over the range of distances D.